/pdf/hardy-interpolating-sequences-of-hyperplanes-2ebfdj9ww2.pdf

Hardy interpolating sequences of hyperplanes

Approximation on a disk II

In this article we study the function algebra generated by z 2 and g 2 on a small closed disk centred at the origin of the complex plane. We prove, using a biholomorphic change of coordinates and already developed techniques in this area, that for a large class of functions g this algebra consists of all continuous functions on the disk.

/pdf/more-approximation-on-disks-2jesrfhtst.pdf

More approximation on disks

Let K be a compact subset of C. The polynomially convex hull of K is defined as K := {z ∈ C : |p(z)| ≤ supK |p| ∀ p ∈ C[z1, . . . , zn]}. The compact set K is said to be polynomially convex if K = K. A closed subset E ⊂ C is said to be locally polynomially convex at z ∈ E if there exists a closed ball Bz centred at z such that E∩Bz is polynomially convex. The aim of this thesis is to derive easily checkable conditions to detect polynomial convexity in certain classes of sets in C, n ≥ 2. This thesis begins with the basic question: Let S1 and S2 be two smooth, totallyreal surfaces in C that contain the origin. If the union of their tangent planes is locally polynomially convex at the origin, then is S1∪S2 locally polynomially convex at the origin? If T0S1 ∩ T0S2 = {0}, then it is a folk result that the answer is, “Yes.” We discuss an obstruction to the presumed proof, and use a different approach to provide a proof. When dimR(T0S1 ∩ T0S2) = 1, it turns out that the positioning of the complexification of T0S1∩T0S2 controls the outcome in many situations. In general, however, local polynomial convexity of S1 ∪ S2 also depends on the degeneracy of the contact of T0Sj with Sj at 0, j = 1, 2. We establish a result showing this. Next, we consider a generalization of Weinstock’s theorem for more than two totallyreal planes in C. Using a characterization, recently found by Florentino, for simultaneous triangularizability over R of real matrices, we present a sufficient condition for local polynomial convexity at 0 ∈ C of union of finitely many totally-real planes is C. The next result is motivated by an approximation theorem of Axler and Shields, which says that the uniform algebra on the closed unit disc D generated by z and h — where h is a nowhere-holomorphic harmonic function on D that is continuous up to ∂D — equals C(D). The abstract tools used by Axler and Shields make harmonicity of h an essential condition for their result. We use the concepts of plurisubharmonicity and polynomial convexity to show that, in fact, the same conclusion is reached if h is replaced by h+R, where R is a non-harmonic perturbation whose Laplacian is “small” in a certain sense. Ideas developed for the latter result, especially the role of plurisubharmonicity, lead us to our final result: a characterization for compact patches of smooth, totally-real graphs in C to be polynomially convex.

Exploring polynomial convexity of certain classes of sets

(2003). A Commonwealth Judge at Work: Lord Cooke in the House of Lords and Privy Council. Oxford University Commonwealth Law Journal: Vol. 3, No. 1, pp. 29-50.

A Commonwealth Judge at Work: Lord Cooke in the House of Lords and Privy Council

© Annales de l’institut Fourier, 1990, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

/pdf/enveloppes-polynomiales-d-unions-de-plans-reels-dans-bbb-c-n-37jbbdbs92.pdf

Enveloppes polynomiales d'unions de plans réels dans ${\Bbb C}^n$

© Annales de l’institut Fourier, 1986, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

/pdf/interpolating-sequences-of-complex-hyperplanes-in-the-unit-1y0fww53q9.pdf

Interpolating sequences of complex hyperplanes in the unit ball of ${\mathbb {C}}^n$

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Enveloppes polynomiales d'unions de plans réels dans ${\Bbb C}^n$

Interpolating sequences of complex hyperplanes in the unit ball of ${\mathbb {C}}^n$